My sister's name is Genevieve and we call her Evie. It's not too much of a stretch as at all - the last syllable makes an 'eve' sound. I reckon Guinevere as Eve is more of a stretch, but you could still do it. Guinevere is such a lovely name.
My favourite names with Eve or Eva as nicknames would be Genevieve, Guinevere (now you've mentioned it!), Evangeline, Yvette and Eveline. You could also use Evelyn, Yvonne, Evelia, Evita. There are heaps of possible names.
I think Eve and Eva are great names just by themselves. They don't sound really childish, the way Evie probably would as a full name, and are such beautiful names.